Calc 2.3a
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Transcript

  • 1. 2.3A PRODUCT AND QUOTIENT RULE Goals: To differentiate a product To differentiate a quotient To find derivative of tangent, cotangent, secant and cosecant To find higher-order derivatives
  • 2.
    • In other words,
    • first•derivative of last + last •derivative of first
    • With a southern accent,
    • First de-last + last de-first
    This rule can be extended to three products or more, see page 119 at the bottom of the page!
  • 3. EXAMPLE 1, P. 120 USING THE PRODUCT RULE
    • Derivative of a product CANNOT, in general, be written as product of two derivatives.
    • Find the derivative of h(x) = (2x – 3x 2 )(4x – 5)
    • Using rule:
    • h’(x) = (2x – 3x 2 )(4) + (4x – 5)(2 – 6x)
    • = 8x – 12x 2 + 8x – 24x 2 – 10 + 30x
    • = -36x 2 + 46x – 10
    • Not using rule but foiling first:
    • h(x) = -12x 3 + 23x 2 – 10x which derives to same thing.
  • 4.
    • In many cases you MUST use product rule!
    • Ex 2 p. 120
    • Find the derivative of y = 4x 3 sin x
    Putting in factored form as much as possible allows us to find the places where a horizontal tangent occurs more easily (where y’ = 0)
  • 5.
    • Ex 3 p. 120
    • Find the derivative of y = 3xsinx – 4cosx
    • (identify the products within the function!)
    • Notice the underlined part is applying the product rule
    Use the product rule if __________________________. Use Constant rule if ____________________________. both factors are variable one of the factors is constant.
  • 6. Break out that cowboy accent! Low dehigh – high delow all over low squared! http://www.youtube.com/watch?v=DdV2UZV7AoA
  • 7.
    • Ex 4, p. 121 Using the quotient rule
    • Find the derivative of
    In general, see if numerator factors, and keep denominator in a factored form Note liberal use of parentheses!
  • 8.
    • Warning – pay special attention to the subtraction required in the numerator – more points lost there than anywhere!
    • Sometimes a rewrite is needed here too:
    • Ex. 5 p. 122 Rewriting before differentiating
    • Find an equation of the tangent line to f(x) at (-1, 1)
    • So
    • and tangent line is y = 1
  • 9.  
  • 10. CAUTION: NOT EVERY QUOTIENT DESERVES THE QUOTIENT RULE!!!!
    • Original Function Rewrite Derive Simplify
  • 11. ASSIGNMENT 2.3A
    • p. 126 #2-12 ev, 13-41 eoo, 53,65,69